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By providing students with a highly interactive and hands-on learning experience, robotics promises to inspire a new generation of mathematical learning. This paper aims to review the empirical evidence on the application of robotics in mathematics education and to define future research perspectives of robot-assisted mathematics education. After a systematic search in online database via keyword search and snowballing approach, we analyzed 20 empirical studies on how to teach and learn mathematical knowledge through robotics. The results indicate that (1) most studies were conducted with a small sample size, the largest research groups were elementary school students and secondary school students, most studies used LEGO robots, robots were primarily applied to teach and/or learn graphics, geometry, and algebra, and almost half of the studies taught mathematics by engaging students in game-like interactions with robots; (2) half of the studies adopted a non-experimental research design, and most studies evaluated student performance through observation, test/examination, questionnaires, or verbal interviews; and (3) instructional implications proposed in the 20 papers can be clustered into four themes: human-robot interaction, connections between mathematics and real life, pedagogical suggestions, and facility conditions. The 20 papers suggest that robotics generally plays an active role in mathematics education; however, there are indeed situations in which no significant improvement was found in students' mathematical learning. In view of this, we prospect the future research perspectives of robot-assisted mathematics education and propose that more rigorous intervention studies could be conducted to further explore the integration of robotics and mathematics education. [ABSTRACT FROM AUTHOR]

In this introduction, we set the stage for a collection of papers from the Co-constructing Mathematics Motivation in Primary Education–A Longitudinal Study in Six European Countries Project (MATHMot for short), an international study aiming to identify the factors that shape the development of motivation in mathematics from a comparative perspective in primary education. Students' motivation, performance, academic emotions, and subject-related identity and agency are observed across six countries: Norway, Sweden, Finland, Estonia, Portugal, and Serbia. MATHMot builds on the belief that one of the main goals of mathematics teaching should be children's long-term motivation for learning the subject, which in turn supports students in striving for exemplary achievement in mathematics. This special section attempts to observe students' mathematics motivation in early grades and how different contributions from the classroom, home or the student's individual characteristics shed light on its development and adjacent concepts like academic emotions and math-related identity and agency. [ABSTRACT FROM AUTHOR]

In Victoria, Australia, senior secondary mathematics students are expected to use technology and thus need to make decisions about using pen-and-paper (P&P) or technology when solving mathematics problems. The predominant technology is a Computer Algebra System (CAS). This study investigated the beliefs about CAS held by twelve Year 11 students as they learnt to use CAS and whether these beliefs were stable over time. These students held a range of beliefs related to the usefulness of CAS, speed of CAS compared to P&P, whether CAS is proper mathematics, choice of CAS or P&P, ease of use, the correctness of answers and solving problems in Mathematical Methods (i.e. the mathematics subject studied). Beliefs are often described as being stable (e.g. McLeod, 1992), but some researchers stress stability needs to be determined empirically rather than being seen as a characteristic of beliefs (e.g. Liljedahl et al., 2012). For this sample of students, stability (rather than instability) is a feature of students' beliefs about CAS. [ABSTRACT FROM AUTHOR]

This study uses ethnic capital theory to explore access to secondary mathematics education among linguistically diverse (LD) migrants in metropolitan and regional New South Wales, Australia. Administrative data from over 50,000 students who completed their Higher School Certificate in 2017 were analysed using multilevel logit regressions and marginal effects. The results indicate that, in Greater Sydney, all linguistically diverse first-generation youth took mathematics courses at higher rates than their peers. So did second-generation migrants from Asian backgrounds. Furthermore, considerably larger proportions of students who spoke East Asian, Indo-Aryan, or Arabic languages studied advanced mathematics. Even when only parents spoke these languages at home, their Australian-born children took advanced mathematics more often. Yet, these second-generation students were less overrepresented than those fluent in parental languages. The paper discusses the potential consequences of LD migrant concentration in Greater Sydney, stressing the importance of equitable mathematics education in metropolitan and regional areas. [ABSTRACT FROM AUTHOR]

Given the increasing prevalence of web technology, web-based mathematics environments have been increasingly widely used in mathematics education for the past two decades. The COVID-19 pandemic has led to an urgent transition from traditional mathematics instruction (TMI) to web-based mathematics instruction (WBMI) at all levels of mathematics education. At this point, it is crucial to scrutinize the effects of WBMI on K-16 students' mathematics learning comprehensively. This meta-analysis research contained a total of 63 studies with 115 effect sizes, which aimed to investigate the effectiveness of WBMI on K-16 students' mathematics learning by incorporating potential moderators, namely mathematics topics, mathematical content standards, feedback status, type of instructional features, age (i.e., grade level), and assessment methods. Based on findings, WBMI has a significantly strong effect on K-16 students' mathematics learning (g = 1.10, p = 0.01, 95% CI [0.95, 1.27]). Moderator analyses reveal that the effect sizes of WBMI on K-16 students' mathematics learning varied significantly depending on all these potential moderators. Additionally, higher-level mathematical concepts, statistics and probability, WBMI with providing feedback, tutorial systems, undergraduate students, and traditional paper-pencil assessment are the strongest moderators in their context. The most notable results of this research are that WBMI is significantly more effective on students' mathematics learning than TMI, while even in the context of WBMI, traditional paper-pencil assessment is significantly more effective than online assessment. This meta-analytic research provides a comprehensive and up-to-date perspective on the effectiveness of WBMI on K-16 students' mathematics learning. [ABSTRACT FROM AUTHOR]

Proving and refuting are fundamental aspects of mathematical practice that are intertwined in mathematical activity in which conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. One aspect underexplored in the education literature is the connection between this activity and the construction by students of knowledge, such as mathematical concepts and theorems, that is new to them. This issue is significant to seeking a better integration of mathematical practice and content, emphasised in curricula in several countries. In this paper, we address this issue by exploring how students generate mathematical knowledge through discovering and handling refutations. We first explicate a model depicting the generation of mathematical knowledge through heuristic refutation (revising conjectures/proofs through discovering and addressing counterexamples) and draw on a model representing different types of abductive reasoning. We employed both models, together with the literature on the teachers' role in orchestrating whole-class discussion, to analyse a series of classroom lessons involving secondary school students (aged 14–15 years, Grade 9). Our analysis uncovers the process by which the students discovered a counterexample invalidating their proof and then worked via creative abduction where a certain theorem was produced to cope with the counterexample. The paper highlights the roles played by the teacher in supporting the students' work and the importance of careful task design. One implication is better insight into the form of activity in which students learn mathematical content while engaging in mathematical practice. [ABSTRACT FROM AUTHOR]

Based on the expectancy-value perspective on identity and identity formation, this paper explores the relationship between math identity (MI) and the dimensions of motivation (i.e. intrinsic value, attainment value, utility value and perceived competence) and math achievement in primary school. An additional aim of our research was to explore these relationships in different cultural contexts and investigate potential gender and grade differences concerning MI. The participants were 11,782 primary school students from Norway, Sweden, Estonia, Finland, Portugal and Serbia. All predictors from the motivation spectrum were significant for students' MI across the examined countries and had a stronger association with MI than math achievement. Among the motivational dimensions, intrinsic value had the strongest association with students' MI. Boys had significantly more positive math identities than girls in Estonia, Finland, Norway and Portugal. The results showed that the grade 4 students perceived themselves less as "math persons" than their grade 3 peers in all countries. [ABSTRACT FROM AUTHOR]

The paper discusses an example of an educational project in modern mathematics in which school students create mathematics that is new to them. The mathematical results produced by the students in the theory of definability also have an "absolute" novelty, i.e., are the basis for professional publications. The described course was based on recent results of this article's authors in the theory of definability. The new results were obtained by a group of 10 schoolchildren from different regions of Russia. [ABSTRACT FROM AUTHOR]

This paper considers an approach to mathematical education adequate to the task of developing mathematics and its applications in the 21th century. This approach is based on improving the efficiency of the educational process by maintaining the motivation of students of various categories. The basis for the formation of motivation is, on the one hand, independent design; invention of mathematical objects, methods of action, and models of the world around us; and the discovery of facts of mathematical reality and, on the other hand, solving of new, unexpected, and feasible tasks for the student. The student's work is similar to the work of a mathematician–researcher and programmer. The possibilities of research work in educational mathematics are significantly expanded due to computer-based intramathematic experiments. Debugging a computer program is a special kind of mathematical experiment. [ABSTRACT FROM AUTHOR]

In this commentary, I draw exclusively on the ten papers in this issue of ZDM focused on Mathematical Tasks and the Student. These papers represent various cultural and theoretical stances and when reading them I was looking for common themes across some or all the papers. Based on my readings, I argue that (a) each task has the potential to promote domain-specific as well as domain-transcendent goals potentially communicating a more coherent and connected picture of mathematics, (b) tasks cannot be considered independently from their enactment and prior enactments should inform future implementations, and (c) the context of the implementation of a task (i.e., the teacher, classroom norms, and student agency) is essential and needs to be considered in the design phase. [ABSTRACT FROM AUTHOR]

In this paper a part of the data from a larger study, the Enactment Project, is explored. The exploration is guided by two research questions. The first is specific to the instructional core of teacher practice while the second is related to student perceptions of how they learn mathematics. The project adopted the Complementary Accounts Methodology. Micro-analysis of how three teachers enacted their instructional objectives, using qualitative data analysis methods, led to the identification of three components of an instructional core, namely, Development of concepts and skills [D], Student work [S] and Review of student work [R] that pervade the lessons of the three teachers. They enacted their instructional objectives through instructional cycles comprising combinations of D, S, and R. A lesson was found to comprise one or more instructional cycles. The post-lesson video-stimulated student interview data, showed that teachers helped their students learn mathematics in their lessons specifically in the S and R activity segments. This paper also dispels the myth about stereotypical mathematics instruction in Asian classrooms whilst contributing to this issue of ZDM in memory of David Clarke. It was his work related to the Complementary Accounts Methodology that provided colleagues in Singapore with a tool to illuminate mathematics instruction in their schools. [ABSTRACT FROM AUTHOR]

This paper is a commentary on the theoretical formulations of the five empirical papers in this special issue. All five papers use aspects of the theory of commognition as presented by Anna Sfard; however, even when the same notions (e.g., rituals or explorations) are incorporated into theoretical frameworks undergirding the research, these notions are not always used in the same way. The analysis of how commognitive theory is used in the papers suggests that it is broad enough to be a useful theoretical lens in diverse settings. However, there are many aspects of Sfard's careful theoretical formulation that are not touched upon, suggesting that there is much more potential for use of the theory than has been realized in these papers. [ABSTRACT FROM AUTHOR]

There is a dearth of research comparing the effects of iPad technology and paper-and-pencil-delivered interventions on student mathematics outcomes. Nine studies have compared intervention modalities such as computer-mediated instruction and teacher-mediated instruction to examine differences in student performance, but only two of these have used iPad technology to do so and neither examined student fact fluency as the outcome. The purpose of this study was to examine the effects of an iPad-based versus a paper-and-pencil-based flashcard intervention on the basic fact fluency of four second-grade boys within a Midwestern US charter school. Using an adapted alternating treatments design, three conditions were compared: iPad-delivered flashcards, paper-and-pencil-based flashcards, and a control condition. The results suggest that for three of four students, there was no difference in gains between treatment conditions, and both were more successful than the no-treatment control. Descriptive data on the acceptability and number of opportunities to respond between intervention modalities are described. [ABSTRACT FROM AUTHOR]

This paper provides on-the-ground accounts of epistemic violence against Black immigrant children in mathematics classrooms. From a critical feminist perspective, we introduce Dotson's notion of silencing as an enactment of epistemic violence. According to Dotson, one way to enact epistemic violence is to damage a particular group's ability to speak and be heard. A successful act of communication depends on the audience's willingness and ability to "hear" the speaker. Therefore, denying this reciprocity in communication is a form of epistemic violence. Using this conceptualization, we conducted a secondary data analysis from a larger study aimed at enhancing teachers' knowledge and abilities to implement problem-solving teaching. We identify and characterize three practices of silencing Black immigrant students in Chilean mathematics classrooms that damage their agency as knowers and doers of mathematics. Beyond language issues, we show that silencing is a form of anti-Black onto-epistemic violence that prevents Black immigrant students from being recognized as legitimate subjects of knowledge in mathematics classrooms. [ABSTRACT FROM AUTHOR]

Representations are key to mathematical activities and meaning-making processes as they are part of modeling, connecting, communicating, and understanding mathematical ideas and concepts. The current study sought to examine a group of novice algebra learners' interactions with different representations from an onto-semiotic approach. A case study method was employed to understand how different algebraic practices (abstracting, generalizing, justifying, and operating on symbols) and functional thinking types (recursive, covariational, and correspondence) were facilitated through working with multiple representations. Three 6th graders participated in the study by completing 12 algebra tasks and taking part in two interviews. The onto-semiotic approach guided the data analysis process that involved the identification of mathematical objects that emerged in the participating students' mathematical practices. Then, the configuration of objects and semiotic functions established by the students in the functional situations was examined to understand the role of representations in the students' development of algebraic thinking and practices. Findings showed that abstraction is an essential process for generalization. Thinking about far figures facilitated abstraction and generalization through helping students construct non-ostensive concrete/pictorial representations. Verbal representations interacted with all representations and preceded symbolic representations. Working with near figures promoted recursive and covariational thinking while examining the far figures usually resulted in correspondence thinking. Implications for the school curriculum are discussed in the paper. [ABSTRACT FROM AUTHOR]

This paper provides an empirical exploration of mathematics teachers' planned practices. Specifically, it explores the practice of foreshadowing, which was one of Wasserman's (2015) four mathematical teaching practices. The study analyzed n = 16 lessons that were planned by pairs of highly qualified and experienced secondary mathematics teachers, as well as the dialogue that transpired, to identify the considerations the teachers made during this planning process. The paper provides empirical evidence that teachers engage in foreshadowing as they plan lessons, and it exemplifies four ways teachers engaged in this practice: foreshadowing concepts, foreshadowing techniques, foregrounding concepts, and foregrounding techniques. Implications for mathematics teacher education are discussed. [ABSTRACT FROM AUTHOR]

The growing popularity of data heterogeneity motivates people to identify homogeneous subgroups with identical parameters. Meanwhile, in many fields of recent data science for some applications, such as personalized education and personalized marketing, the massive data are usually recorded as categorical or ordinal variables, which highlights the importance of performing subgroup analysis on those ordinal outcomes. In this paper, we propose a cumulative link model with subject-specific intercepts to detect and identify homogeneous subgroups through concave pairwise fusion penalty for ordinal response, where heterogeneity arises from some unknown or unobserved latent factors. The concave fusion method can simultaneously determine the number of subgroups, identify the group membership, and estimate the regression coefficients. An alternating direction method of multipliers algorithm with concave penalties for the generalized linear regression model with logit link is developed and its convergence property is studied. We also establish the oracle property of the proposed penalized estimator under some mild conditions. Our simulation studies show that the proposed method could recover the heterogeneous subgroup structure effectively when the response of interest is ordinal. Further, the advantages of our method are illustrated by the analysis on a Mathematics Student Performance Data Set of two public schools from the Alentejo region of Portugal. [ABSTRACT FROM AUTHOR]

Recent research syntheses have identified several potentially high-leverage teaching strategies for improving low-attaining secondary school students' learning of mathematics. These strategies include the structured use of representations and manipulatives and an emphasis on derived facts and estimation. This paper reports on 70 semi-structured interviews conducted with low-attaining students in Years 9–10 (ages 13–15) in England. The interviews addressed the students' perceptions of learning mathematics and the teaching strategies that they experienced and believed were most helpful. Many students reported rarely using number lines, not spontaneously estimating answers and being unfamiliar with derived facts. During the interviews, with minimal direction, students often showed that they were well able to make use of these strategies; however, they did not report making spontaneous use of them independently. We conclude that many of the most well-evidenced and recommended strategies to support low-attaining students in mathematics appear to be unfamiliar and unvalued, and we discuss how this might be addressed. [ABSTRACT FROM AUTHOR]

This paper attempts to deepen our understanding of teachers' selection and modification of mathematical tasks for their classroom instruction. In this paper, we focus on teachers' orientations and resources that subtly influence teachers' choices during their routine undertaking of task selection and modification. Interviews using stimulus material were used to induce teachers to expound their thought process as they deliberated the suitability of tasks and made choices related to task selection and modification. These thought processes are critical as it provides potential explanation for teachers' task inclinations. The findings of this study show that whilst beliefs of the teaching and learning of mathematics and beliefs of students' capabilities and capacities are preponderant as the teachers deliberate on which tasks they would use, the orientations within these beliefs differ amongst them. These orientations, productive or unproductive, influence their decisions during task selection and modification. Teacher resource specific to mathematical knowledge for teaching (MKT) is also critical in impacting teachers' choice of tasks. Teacher orientations and expertise impact the range and extent the teachers draw on their MKT, resulting in dissimilar inclinations towards tasks for mathematics instruction. In addition, resources such as preparation time, curriculum time and materials teachers have at hand influence their choices when selecting and modifying mathematical tasks for instruction. [ABSTRACT FROM AUTHOR]

Properties of arithmetic operations are important in mathematics and play a significant role in elementary school students' learning of mathematics. At the same time, it is quite challenging to help students learn arithmetic operational properties beyond memorization, so that they know when and how to use a specific arithmetic property. This paper reports a study containing two parts to explore students' learning of the Commutative Property of Addition (CPA) as related to the content presentation in textbooks. Part one is to assess selected students' performance on CPA at the end of Grade two, and part two is to explore possible contributions of textbooks to students' performance in Shanghai. Firstly, we evaluated the performance of 41 second-graders on CPA by collecting data through a paper-and-pencil test, which was followed up with individual interviews. Through analyzing how textbooks present and organize this topic, we sought to better understand from a curriculum standpoint how students' performance on CPA is impacted by textbooks. Results revealed that Shanghai textbooks make a positive contribution to students' learning performance. The findings from this study help to shed light on how those textbooks aid student learning, and provide an example for further reflection on student learning of arithmetic operational properties in different curricula. [ABSTRACT FROM AUTHOR]

Background: There is anecdotal evidence that many elementary teachers integrate mathematics lessons and art activities by having students first make colorful, rich material that is subsequently used in an instructional activity. However, it is unclear whether such activities effectively promote learning and transfer of mathematical concepts. The goal of the present research was to examine the use and effectiveness of such "math-and-art" activities on children's ability to acquire basic fraction knowledge. We report the results of a survey of practicing elementary school teachers in the United States, their use of activities involving physical material, and the resources they use for ideas to supplement the standard curriculum. Two experiments examined first-grade students' learning, transfer, and recognition of fraction knowledge from rich, contextualized material versus simple, generic material. Results: The survey results confirm that many U.S. teachers use math-and-art activities and are often inspired by informal sources, such as Pinterest and YouTube. Experiment 1 examined the effectiveness of colorful, contextualized student-constructed material (paper pizzas) versus simple, pre-made material (monochromatic paper circles) in an instructional activity on fractions. Students who used the pre-made circles scored higher than those who used the student-made pizzas on pre-instruction tests of basic fraction knowledge, immediate tests of learning, and delayed tests of transfer. Experiment 2 tested students' ability to spontaneously write fractions to describe proportions of pizzas and circles. Students who answered generic circle questions first were markedly more accurate than those who answered pizza questions first. Conclusions: These findings suggest that rich, contextualized representations, including those made by the student, can hinder students' learning and transfer of mathematical concepts. We are not suggesting that teachers never integrate mathematics and colorful, contextualized material, and activities. We do suggest that elementary students' mathematics learning can benefit when initial instruction involves simple, generic, pre-made material and opportunities for students to make and use colorful, contextualized representations come later. [ABSTRACT FROM AUTHOR]

It is widely believed that a joint factor analysis of item responses and response time (RT) may yield more precise ability scores that are conventionally predicted from responses only. For this purpose, a simple-structure factor model is often preferred as it only requires specifying an additional measurement model for item-level RT while leaving the original item response theory (IRT) model for responses intact. The added speed factor indicated by item-level RT correlates with the ability factor in the IRT model, allowing RT data to carry additional information about respondents' ability. However, parametric simple-structure factor models are often restrictive and fit poorly to empirical data, which prompts under-confidence in the suitablity of a simple factor structure. In the present paper, we analyze the 2015 Programme for International Student Assessment mathematics data using a semiparametric simple-structure model. We conclude that a simple factor structure attains a decent fit after further parametric assumptions in the measurement model are sufficiently relaxed. Furthermore, our semiparametric model implies that the association between latent ability and speed/slowness is strong in the population, but the form of association is nonlinear. It follows that scoring based on the fitted model can substantially improve the precision of ability scores. [ABSTRACT FROM AUTHOR]

Today's adolescents are digital natives. Information and knowledge are abundant and easily accessible. Learning a subject perceived to be difficult such as mathematics may present a relatively high cost and emotional burden. This study investigated the relations among cost, academic emotion, and performance in mathematics. A longitudinal survey of two waves was conducted among 754 (52.12% males) seventh grade Taiwanese students at both Time 1 and Time 2 with an average age of 12.59 years (SD =0.53) at Time 1. Mathematics academic emotions, cost, and academic performance were significantly correlated. Cross-lagged analysis revealed that mathematics academic emotions (MAE) predicted students' mathematics performance but indirectly through perceived cost. Academic performance in mathematics was positively associated with subsequent mathematics academic emotions. A loop was detected between MAE and cost as well as cost and academic performance. This paper clarifies the interrelationships among these variables and reveals the crucial role of the perceived cost. Activities that aim to reduce students' perceived cost of engaging in mathematics tasks will improve their learning emotions and mathematics achievement. [ABSTRACT FROM AUTHOR]

Problems for students with a high mathematical potential can be far more complex and challenging than problems normally used in regular classrooms. The problems we developed for mathematically gifted students offer the possibility of acquiring and using heuristics (so called patterns of actions) which are useful in problem solving processes. Because these kinds of capablities are valuable for all students, we ask whether it is possible to work on the problems in regular classrooms. In this paper two studies are presented, which examine to what extent students in regular classrooms are capable of working on problems designed for mathematically gifted students. The results show that the problems used in the support programme can also be successfully applied in regular classroom teaching and that students working on them experience high motivation. However, students of our support programme achieved better results regarding generalization and proving. In addition, they needed less time to solve the problems. [ABSTRACT FROM AUTHOR]

A basic mental model (BMM—in German 'Grundvorstellung') of a mathematical concept is a content-related interpretation that gives meaning to this concept. This paper defines normative and individual BMMs and concretizes them using the integral as an example. Four BMMs are developed about the concept of definite integral, sometimes used in specific teaching approaches: the BMMs of area, reconstruction, average, and accumulation. Based on theoretical work, in this paper we ask how these BMMs could be identified empirically. A test instrument was developed, piloted, validated and applied with 428 students in first-year mathematics courses. The test results show that the four normative BMMs of the integral can be detected and separated empirically. Moreover, the results allow a comparison of the existing individual BMMs and the requested normative BMMs. Consequences for future developments are discussed. [ABSTRACT FROM AUTHOR]

This paper focuses on the importance of covariational reasoning within the processes of mathematics teaching and learning. Despite the internationally recognized relevance of covariation, research shows that only a small percentage of students and teachers is able to adopt covariational reasoning and the majority of mathematics curricula do not contain explicit references to covariational skills. In particular, when covariational reasoning manifests as conceptual knowledge, it becomes challenging to assess, and the need for innovative methods of assessment emerges; there is a need for suitable assessment to highlight the characteristics of covariation and capture the various features that characterize conceptual understanding. Comparative judgement (CJ) is an innovative assessment method based on collective expert judgements of students' work rather than requiring scoring rubrics. Due to its holistic approach, CJ is particularly suitable for assessing complex mathematical competencies, and, as we shall see in this study, it proved to be appropriate for the covariation's assessment. In details, our study aims to investigate the perception and relevance attributed by mathematics teachers to covariation as a theoretical construct and the way CJ can help in the assessment of covariational reasoning skills underlying a less structured modelling task. [ABSTRACT FROM AUTHOR]

Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students' self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students' self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students' self-efficacy across a sequence of 3–4 lessons when students were learning a new topic in mathematics (nStudents = 170, nTime-points = 596). Nine classes of Norwegian grade 6 (n = 77) and grade 10 students (n = 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students' self-efficacy across lesson sequences, (b) differences in the starting point and change of students' self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed. [ABSTRACT FROM AUTHOR]

Offering explanations is a central part of teaching mathematics, and understanding those explanations is a vital activity for learners. Given this, it is natural to ask what makes a good mathematical explanation. This question has received surprisingly little attention in the mathematics education literature, perhaps because the field has no agreed method by which explanation quality can be reliably assessed. In this paper, we explore this issue by asking whether mathematicians and undergraduates agree with each other about explanation quality. A corpus of 10 explanations produced by 10 mathematicians was used. Using a comparative judgement method, we analysed 320 paired comparisons from 16 mathematicians and 320 from 32 undergraduate students. We found that both mathematicians and undergraduates were able to reliably assess the quality of a set of mathematical explanations. Furthermore, the assessments were largely consistent across the two groups. Implications for theories of mathematical explanation are discussed. We conclude by arguing that comparative judgement is a promising technique for exploring explanation quality. [ABSTRACT FROM AUTHOR]

Confidence assessment (CA) involves students stating alongside each of their answers a confidence rating (e.g. 0 low to 10 high) to express how certain they are that their answer is correct. Each student's score is calculated as the sum of the confidence ratings on the items that they answered correctly, minus the sum of the confidence ratings on the items that they answered incorrectly; this scoring system is designed to incentivize students to give truthful confidence ratings. Previous research found that secondary-school mathematics students readily understood the negative-marking feature of a CA instrument used during one lesson, and that they were generally positive about the CA approach. This paper reports on a quasi-experimental trial of CA in four secondary-school mathematics lessons (N = 475 students) across time periods ranging from 3 weeks up to one academic year, compared to business-as-usual controls. A meta-analysis of the effect sizes across the four schools gave an aggregated Cohen's d of –0.02 [95% CI –0.22, 0.19] and an overall Bayes Factor B01 of 8.48. This indicated substantial evidence for the null hypothesis that there was no difference between the attainment gains of the intervention group and the control group, relative to the alternative hypothesis that the gains were different. I conclude that incorporating confidence assessment into low-stakes classroom mathematics formative assessments does not appear to be detrimental to students' attainment, and I suggest reasons why a clear positive outcome was not obtained. [ABSTRACT FROM AUTHOR]

In this paper, we identify various forms of geometric work carried out by student teachers who were asked to perform a geometric task for the estimation of a land area. The theory of Mathematical Working Spaces is used to analyze and characterize the work produced. This study provides evidence that students developed forms of geometric work that are compliant with at least two distinct geometric paradigms, one characterized by the utilization of measuring and drawing tools and the other by a property-based discourse on proof. Significantly, a sizable number of students also developed work forms that do not correspond to any geometrical paradigm. A broader purpose of this paper is to highlight three criteria born by the theory and shown to be useful for the description and evaluation of geometric work: compliance, completeness, and correctness. [ABSTRACT FROM AUTHOR]

Although there is much research exploring students' covariational reasoning, there is less research exploring the ways students can leverage such reasoning to coordinate more than two quantities. In this paper, we describe a system of covariational relationships as a comprehensive image of how two varying quantities, having the same attribute across different objects, each covary with respect to a third quantity and in relation to each other. We first describe relevant theoretical constructs, including reasoning covariationally to construct relationships between quantities and reasoning covariationally to compare quantities. We then present a conceptual analysis entailing three interrelated activities we conjectured could support students in reasoning covariationally to conceive of a system of covariational relationships and represent the system graphically. We provide results from two teaching experiments with four middle school students engaging in tasks designed with our conceptual analysis in mind. We highlight two different ways students reasoned covariationally compatible with our conceptual analysis. We discuss the implications of our results and provide areas for future research. [ABSTRACT FROM AUTHOR]

Disparate approaches to assessment pursue differing purposes. Clinical assessment interviews like the Australian early numeracy interview (ENI; Clarke et al. 2002) or its German version ElementarMathematisches BasisInterview (EMBI; Peter-Koop et al. 2007) can be used to gain insights into children's mathematical thinking and aim at providing a basis for instruction and intervention. In practical trials in inclusive classrooms, the EMBI proves as suitable for the use in inclusive mathematics education. However, this interview reaches its limits in the case of sensory impairments such as blindness or severe visual impairment. This paper presents a modified version of the EMBI for children with visual impairments and the analysis of respective case study data. In addition to the description of mathematical abilities and understanding in terms of growth points, a more sophisticated competence grid is used for a more detailed analysis of children's competencies. Both the adaptation of the interview and the competence grid are critically reviewed in this paper. [ABSTRACT FROM AUTHOR]

This paper presents findings from a case study on the impact of high stakes oral performance assessment on third year mathematics students' approaches to learning (Entwistle and Ramsden, Understanding student learning, 1983). We choose oral performance assessment as this mode of assessment differs substantially from written exams for its dialogic nature and because variation of assessment methods is seen to be very important in an otherwise very uniform assessment diet. We found that students perceived the assessment to require conceptual understanding over memory and were more likely to employ revision strategies conducive to deep learning (akin to conceptual understanding) when preparing for the oral performance assessment than when preparing for a written exam. Moreover, they reported to have engaged and interacted in lectures more than they would have otherwise, another characteristic conducive to deep learning approaches. We conclude by suggesting some implications for the summative assessment of mathematics at university level. [ABSTRACT FROM AUTHOR]

During the last decade an eruptive increase in the demand for intelligent m-learning environments has been observed since instructors in the online academic procedures need to ensure reliability. The research for decision systems seemed inevitable for flexible and effective learning in all levels of education. The prediction of the performance of students during their final exams is considered as a difficult task. In this paper, an application is presented, contributing to an accurate prediction which would assist educators and learning experts in the extraction of useful knowledge for designing learning interventions with enhanced outcomes. [ABSTRACT FROM AUTHOR]

The notion of definition is a central building block of mathematics. In addition to specific definitions, it is desirable that students learn about meta-mathematical aspects of definitions such as their role of classification, and about characteristics of definitions such as arbitrariness. We investigated how such meta-mathematical aspects emerge when students are led to construct a definition such as point of inflection or tangent line, and how the constructing process interacts with meta-mathematical aspects. We used Abstraction in Context as theoretical framework because of its suitability to analyze processes of knowledge construction. We found that meta-mathematical aspects of definitions influence and are influenced by students' processes of constructing definitions. We argue that awareness of meta-mathematical aspects improves students' conception of mathematical definitions. [ABSTRACT FROM AUTHOR]

This paper focuses on the discourse of teachers and instructors in a TEAMS (Teaching Exploratively for All Mathematics Students) professional development (PD) setting and on the discursive mechanisms that afford such learning. We conceptualize learning in this PD as a change in a teacher's pedagogical discourse—from alignment with Delivery Pedagogical Discourse (DPD) to alignment with Explorative Pedagogical Discourse (EPD). This change in pedagogical discourse involves commognitive conflicts between the DPD and EPD. Data for the study include 8 videotaped sessions from a 2-year TEAMS PD for middle school mathematics teachers. Results show that conflict revolved around four themes. The most common was about students' abilities and its relation to instruction. The others were: Who is responsible for constructing mathematics knowledge in class? Limited view of the EPD, and what is considered ordinary instruction? We also identified four levels of conflict, ranging from commognitive conflict that is implicit to the participants, to conflict that is made explicit and reflected upon. Finally, results show movement from implicit conflict towards explication of conflict during the two years of the PD. [ABSTRACT FROM AUTHOR]

Teaching effectiveness is a core issue in educational research; however, there is little consensus about the most important results of classroom teaching from an international perspective. The effectiveness of teaching has remained a 'black box' for a long time. In the secondary study described in this paper we used empirical data for Shanghai taken from the international Organization for Economic Co-operation and Development (OECD) study Global Teaching InSights (GTI)—initially the Teaching and Learning International Survey (TALIS) Video Study—which was based on videotaped direct observations of classroom teaching. Eighty-five junior high school mathematics teachers and their students in Shanghai were observed to explore the impact of specific teaching practices on students' interest, self-efficacy, and mathematics achievement scores. The results revealed that social-emotional support and instruction quality were the key dimensions relating to the characteristics and differences of mathematics lessons in Shanghai. While the former had a significantly positive impact on students' general mathematics self-efficacy, the latter had a significantly positive impact on students' mathematics interest. Although specific teaching practices had no significant direct impact on students' mathematics achievement scores, social-emotional support and instruction quality considerably influenced students' academic performance in an indirect way via general self-efficacy. [ABSTRACT FROM AUTHOR]

In this paper, we report on the cross-cultural comparative analysis that was undertaken as part of the Social Unit of Learning project investigating student collaborative problem solving in China and Australia. The video-based research examined the 'negotiative foci' of dyadic student interaction during collaborative problem solving within the mathematics classroom setting. In the analysis we take negotiative focus as the central construct and compare the student interactions in terms of their focus on either facts and procedures, didactical norms, or social/interpersonal considerations. We report on the analysis based on an open-ended pair task (Household task) that was completed by year 7 students (mostly 12- to 13-year olds) in China and Australia. In the study we analysed the video and written responses of 20 student pairs from three year 7 classes in China (40 students in total) and 25 student pairs from two year 7 classes in Australia (50 students in total). The findings from this comparative study provide empirical evidence regarding how the participating students in each country interpreted and negotiated meaning as they tackled the open-ended task collaboratively. [ABSTRACT FROM AUTHOR]

Research exploring undergraduates' tool preferences in tertiary mathematics is currently limited. In this paper, we explore the tools mathematics and engineering students use when studying mathematics and the ways undergraduates combine these tools. Results from our survey (N = 628) showed that students in our sample use mostly their notes, the university's virtual learning environment (VLE), other students and textbooks. Mathematics students were found using more online encyclopaedias, the university's VLE, instant messaging and other students, whereas engineering students reported using more textbooks. By performing a cluster analysis, undergraduates were assigned to five distinct tool-use profiles: the peer-learning group (above average use of other students, instant messaging apps and social media); the online-learning group (above average use of online videos, online encyclopaedias, Wolfram Alpha and the university's VLE); the blended-learning group (above average use of all the tools at their disposal); the no-users group (below average use of all tools except textbooks); and the selective-learning group (above average use of textbooks, online videos and lecturers). The majority of mathematics students were distributed across the peer-learning, online-learning and blended-learning groups, whereas most of the engineering students were found in the blended-learning, no-users and selective-learning groups. [ABSTRACT FROM AUTHOR]

Mathematicians commonly distinguish two modes of work in the discipline: Problem solving, and theory building. Mathematics education offers many opportunities to learn problem solving. This paper explores the possibility, and value, of designing instructional activities that provide supported opportunities for students to learn mathematics theory-building practices. It begins by providing a definition of these theory-building practices on the basis of which to formulate principles for the design of such instructional activities. The paper offers theoretical arguments that theory-building practices serve not only the synthesizing role that they play in disciplinary mathematics, but they also have the potential to enrich learners' reasoning powers and enhance their problem solving skills. Examples of problem sets designed for this purpose are provided and analyzed. [ABSTRACT FROM AUTHOR]

Formulas are involved in most parts of the mathematical curriculum in upper secondary education and in everyday mathematics classrooms, but research shows that students have difficulties using formulas adequately. When students are presented with a task, the task activates a conceptual frame in the students, making them perceive formulas in a specific way, thereby affecting their mathematical behaviour. In this paper, building ideal types of patterns of mathematical behaviours is used to conceptualise 'view on formula' specified by several specific views. The concept of 'view on formula' is applied in an analysis of a classroom episode pointing to reasons for the difficulties students have with handling formulas. When views are either missing or not used in a flexible way this can lead to an unsuccessful handling of formulas. Also, students' views on formula may indicate what knowledge is missing for solving a task involving formulas. Taken together, this points to the importance of paying attention to views on formulas in the everyday mathematics classroom. [ABSTRACT FROM AUTHOR]

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved. [ABSTRACT FROM AUTHOR]

In this paper, we analyse a large, opportunistic dataset of responses (N = 219,826) to online, diagnostic multiple-choice mathematics questions, provided by 6–16-year-old UK school mathematics students (N = 7302). For each response, students were invited to indicate on a 5-point Likert-type scale how confident they were that their response was correct. Using demographic data available from the online platform, we examine the relationships between confidence and facility (the proportion of questions correct), as well as gender, age and socioeconomic disadvantage. We found a positive correlation between student confidence and mean facility, higher confidence for boys than for girls and lower confidence for students classified as socioeconomically disadvantaged, even after accounting for facility. We found that confidence was lower for older students, and this was particularly marked across the primary to secondary school transition. An important feature of the online platform used is that, when students answer a question incorrectly, they are presented with an analogous question about 3 weeks later. We exploited this feature to obtain the first evidence in an authentic school mathematics context for the hypercorrection effect (Butterfield & Metcalfe J EXP PSYCHOL 27:1491–1494, 2001), which is the observation that errors made with higher confidence are more likely to be corrected. These findings have implications for classroom practices that have the potential to support more effective and efficient learning of mathematics. [ABSTRACT FROM AUTHOR]

This paper proposes an innovative statistical method to measure the impact of the class/school on student achievements in multiple subjects. We propose a semiparametric model for a bivariate response variable with random coefficients, that are assumed to follow a discrete distribution with an unknown number of support points, together with an Expectation-Maximization algorithm—called BSPEM algorithm—to estimate its parameters. In the case study, we apply the BSPEM algorithm to data about Italian middle schools, considering students nested within classes, and we identify subpopulations of classes, standing on their effects on student achievements in reading and mathematics. The proposed model is extremely informative in exploring the correlation between multiple class effects, which are typical of the educational production function. The estimated class effects on reading and mathematics student achievements are then explained in terms of various class and school level characteristics selected by means of a LASSO regression. [ABSTRACT FROM AUTHOR]

We consider a home health care routing problem with scarce resources which is a variation of the vehicle routing problem with resource constraints. In this problem, the services are provided by nurses and aids at patients' homes. There are different types of patients that are categorized based on the services they demand and should be serviced by appropriate staff teams or single personnel where the teams are transported by rental vehicles. In this problem, the objective is to minimize the total transportation cost, by considering the patients' requirements and the qualification of the staff, as well as the resource limitations. In this paper, we present a mathematical formulation of the problem that is modeled similar to a vehicle routing problem and propose a branch-and-price algorithm to solve it. Our proposed algorithm utilizes speed up techniques from the literature to enhance the effectiveness of the proposed solution method. A comprehensive computational study is conducted on Solomon based instances and the results illustrate the efficiency of the algorithm and the effective use of available resources. [ABSTRACT FROM AUTHOR]

To help students acquire mathematics and science knowledge and competencies, educators typically use multiple external representations (MERs). There has been considerable interest in examining ways to present, sequence, and combine MERs. One prominent approach is the concreteness fading sequence, which posits that instruction should start with concrete representations and progress stepwise to representations that are more idealized. Various researchers have suggested that concreteness fading is a broadly applicable instructional approach. In this theoretical paper, we conceptually analyze examples of concreteness fading in the domains of mathematics, physics, chemistry, and biology and discuss its generalizability. We frame the analysis by defining and describing MERs and their use in educational settings. Then, we draw from theories of analogical and relational reasoning to scrutinize the possible cognitive processes related to learning with MERs. Our analysis suggests that concreteness fading may not be as generalizable as has been suggested. Two main reasons for this are discussed: (1) the types of representations and the relations between them differ across different domains, and (2) the instructional goals between domains and subsequent roles of the representations vary. [ABSTRACT FROM AUTHOR]

In this paper, we expand our prior work on mathematics education in contexts of language diversity by elaborating on the three perspectives on language described by Ruiz (NABE J 8(2):15-34, ): language-as-right, language-as-resource, and language-as-problem. We illustrate our arguments with data taken from research contexts in Catalonia-Spain and South Africa. In these two parts of the world, the language policy in education has long been an issue, with a monolingual orientation that values one language (i.e., Catalan in Catalonia and English in South Africa) over others. Throughout the introduction of specific examples of policy documents, classroom practices, and participants' reports, our main point is that the right of using the students' languages makes sense because it is itself more than an intrinsic human right; it is an option that potentially benefits the creation of mathematics learning opportunities. Especially for the instances of classroom practices, our examples can be considered as representative in that they point to a common situation in our data: despite the fact of the language of learning and teaching being fixed, there is room for the learners and the teacher to take or react to a decision on what language to use, with whom, and how in concrete moments of the interaction. However, on the basis of our studies and drawing on the literature in mathematics education and language diversity, we argue that language rights are not sufficiently connected to language as a pedagogical resource. The enactment of these rights is still contributing in many ways to the social and political construction of problems concerning the role of certain languages in classroom interaction. We conclude the paper by discussing some possibilities for framing language as a resource that provide effective support to all students' learning of mathematics. [ABSTRACT FROM AUTHOR]

One major obstacle for integrating the history of mathematics in teaching (hereafter referred to as IHT) is how to help teachers, particularly those who lack experience in IHT, use historical materials in their teaching. Based on many IHT practices in Shanghai, this paper proposes an IHT framework including two parts: one is a triangular pyramid IHT model, and the other is a design-based IHT procedure. The pyramid model is composed of three different communities: teachers, researchers, and historians of mathematics. The design-based procedure, also called operating model, includes cyclic stages, which are investigation, design, implementation, assessment, and publication. The primary purpose of this framework is to build a bridge between theory and practice by combining the two models to address particular teaching concerns and improve teaching. In addition, this paper details the process of the dynamic pyramid model and the design-based IHT procedure, and illustrates an example of the application of this framework. Finally, this paper discusses the differences and similarities between this model and others and also discusses the challenge of applying this model in practice. [ABSTRACT FROM AUTHOR]

This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform –which was eventually itself replaced in the longer term – provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms ‘numeration units’ (ones, tens, hundreds, thousands, etc.) and ‘powers-of-ten written in figures’ (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the twentieth century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic. [ABSTRACT FROM AUTHOR]

This paper presents a case study of the experiences of a special educator named Ms. Montes (pseudonym) teaching standards-based mathematics during Emergency Remote Teaching (ERT) during spring 2020. Ms. Montes was interviewed twice during this period; data were analyzed through inductive thematic analysis. Pre-COVID, Ms. Montes provided her students daily opportunities to tackle challenging mathematical problems and taught self-regulation strategies for students to better understand themselves as learners. After the shift to ERT, Ms. Montes described "the wall between us" as various barriers that made teaching mathematics online far more challenging. Challenges included supporting students with productive struggle when not physically present with them and supporting student self-regulation during mathematical problem-solving. Supporting students with disabilities to learn mathematics during ERT and distance learning will require considering emotional and affective dimensions of learning. Coaching students and families in self-regulation strategies could support student engagement in mathematical problem-solving in online learning. [ABSTRACT FROM AUTHOR]